Analysis of the SOD method

Apply formula (6) twice to obtain the top row of the matrix in the propagation equation

$$\left( \begin{array}{c} \psi ^{n+1}\\ \psi ^{n} \end{array}\rig... ...right) \left( \begin{array}{c} \psi ^{n-1}\\ \psi ^{n-2} \end{array}\right) .$$

The eigenvalues of this propagation matrix are

$$\lambda _{1/2} = 1-2\frac{\Delta t^{2}}{\hbar ^{2}}\widehat{H}\pm 2\frac{\Delta t}{\hbar }\widehat{H}\sqrt{\frac{\Delta t^{2}}{\hbar ^{2}}\widehat{H}^{2}-1}$$

$$= 1-2\frac{\Delta t^{2}}{\hbar ^{2}}\widehat{H}\pm 2\frac{\Delta t}... ...}^{2}-\frac{3}{4!}\frac{\Delta t^{4}}{\hbar ^{4}}\widehat{H}^{4}+\cdots \right)$$ (7)

The second expression stems from the Taylor-series of the squareroot at the expansion point $-1$.

The determinant of the propagation matrix has to be unity to make the map area unitary: $\lambda _{1}\lambda _{2}=1.$ The mapping is stable only if the eigenvalues lie on the complex unit circle for otherwise $\lambda _{1}>1$ and the method diverges. So the radicant has to be negative, hence:

$$\Delta t<\frac{\hbar }{E_{max}}.$$

The eigenvalue of the exact operator is

$$\lambda _{exact}=e^{-2\frac{i}{\hbar }E_{m}\Delta t}=1-2i\frac{\D... ...t^{2}}{\hbar ^{2}}E_{m}^{2}-4i\frac{\Delta t^{3}}{3\hbar ^{3}}E_{m}^{3}+\cdots$$

Comparing with (7) gives the error per time-step

$$\varepsilon =\frac{\left( \Delta tE_{m}\right) ^{3}}{3\hbar ^{3}}.$$

In practice, a time-step which is safely smaller than the optimal one is chosen, usually $\Delta t=\frac{\Delta t_{opt}}{5}$ because this yields a high accuracy even after a large number K of recursion steps:

$$\varepsilon \approx \frac{K}{375}.$$

Now, $\left\langle \psi (t)\right\vert$(6)$\rangle$ is

$$\left\langle \psi (t)\vert \psi (t+\Delta t)\right\rangle =\left\... ...hbar }\Delta t\left\langle \psi (t)\vert \widehat{H}\vert \psi (t)\right\rangle$$ (8)

and $\langle$(6) $\left\vert \psi (t)\right\rangle$ is

$$\left\langle \psi (t+\Delta t)\vert \psi (t)\right\rangle =\left\... ...ar }\Delta t\left\langle \psi (t)\vert \widehat{H}\vert \psi (t)\right\rangle .$$ (9)

Adding these two equations, ((8)+(9)), yields

$$Re\left\langle \psi (t+\Delta t)\vert \psi (t)\right\rangle =Re\left\langle \psi (t)\vert \psi (t-\Delta t)\right\rangle =const$$

which means norm conservation for real overlaps. Similarly, $\left\langle \widehat{H}\psi (t)\right\vert$can be multiplied with $\vert(\ref{SOD})\rangle$ to obtain energy conservation.